Journal of Archaeological Method and Theory

, Volume 22, Issue 1, pp 306–344 | Cite as

The Diffusion of Fired Bricks in Hellenistic Europe: A Similarity Network Analysis

Article

Abstract

In this study, similarity networks are used to analyse the diffusion of fired bricks across Hellenistic Europe, initiated in the north Aegean in the fourth century BCE. In a similarity network, archaeological contexts with similar attributes are connected according to a predefined criterion, and the edges are interpreted as potential social or causal links. Here we allow any kind of similarity relation between brick contexts as a criterion for connection, where different kinds of attributes can be mixed. The analysis suggests that there was indeed a diffusion process, as opposed to random, independent appearances of fired bricks. This diffusion seems to have taken place in a small-world social network of builders and decision-makers, where the range of most connections was shorter than 250 km but where long-range connections were also present. The data seemingly exclude a scale-free network, in which the diffusion is governed by a few dominant hubs. An early cluster of homogeneous brick usage appears in the south-eastern part of Hellenistic Europe, as well as a late cluster in the north-west. In the late cluster, the homogenisation applies to larger constructions (public or military) than in the early cluster (mainly sepulchral), suggesting that the use of fired bricks became established at higher societal levels in the last century BCE.

Keywords

Similarity networks Networks Diffusion of innovations Fired bricks Hellenistic period 

Introduction

In this paper, network analysis of the archaeological record is used to infer characteristics of the spread of fired bricks across the Mediterranean during the Hellenistic period. This process is most clearly visible in the European part of the region and is an example of the diffusion of innovations.

The Network Approach

The methodological framework is that of general similarity network analysis (Östborn and Gerding 2014). In this approach, all kinds of similarities (including well-defined differences) that relate the attributes of two archaeological contexts are allowed as criteria to connect them into a network. Different kinds of attributes can be mixed in a single criterion for connection. Since we are interested in a spatial process of diffusion, we focus on networks of geographically located brick contexts rather than networks of their attributes.

The basic idea is that more similar contexts are more probably causally related than are less similar contexts. To detect such causal relations effectively in the material record, it is not enough to consider similarities in a single pre-chosen attribute. Several attributes should be included in the analysis, and several similarity criteria for connection should be tried.

Such exploratory analysis is ideally complemented with statistical analysis of the observed network patterns. Only statistically significant patterns should be used as a basis for historical interpretation. We refer the reader to the paper cited above for a broader motivation of the approach, and of the choice of statistical methods.

Of course, it is not a new idea to reconstruct social or causal networks from similarity networks. Reviews are found in Brughmans (2010; 2013) and Östborn and Gerding (2014). Good surveys of archaeological network analysis are also found in Bentley and Maschner (2003), Knappett (2011, 2013) and Malkin et al. (2009).

Hellenistic Fired Bricks

Fired bricks were manufactured continuously in Mesopotamia from at least the beginning of the third millennium BCE (Campbell 2003; Moorey 1994). In the reign of Nebuchadnezzar, brick production reached its peak. The fact that Babylon’s outer city wall was made entirely of fired bricks became a popular topos in Greek and Roman literature (Gerding forthcoming). Herodotus (1.179) described it in the mid-fifth century BCE, and Aristophanes alluded to it in one of his plays (Aves 552).

However, there is no evidence for the use of fired bricks for masonry in the Graeco-Roman world in the Archaic or Classical periods. The westernmost example of the Near Eastern brick tradition is the city wall of Sardis, erected in the last quarter of the seventh century BCE. But the trail seems to stop here. A considerable chronological lacuna speaks against a link between this find and the Hellenistic fired bricks in Greece. The use of fired bricks in Europe should therefore be regarded as a Hellenistic innovation.

Known finds indicate that fired bricks were first introduced in Europe in the north Aegean in the mid-fourth century BCE. Within a few decades after their first appearance, they had spread to numerous other places. For a long time, however, the use of fired bricks was limited and sporadic, and it took more than 300 years before the new building material became widely adopted in this part of the world.

In the early first century CE, standardised fired bricks were being produced in large quantities around Rome. From here, the innovation spread rapidly all over the Roman Empire and gained an important role in Roman architecture, thereby significantly affecting building traditions in later periods as well. Consequently, different aspects of Roman bricks have been thoroughly studied throughout the nineteenth and twentieth centuries (e.g. Adam 1994; Blake 1947, 1959; Bloch 1947; Boëthius and Ward-Perkins 1970; Brodribb 1987; Choisy 1873; Cozzo 1936; Durm 1885; Lugli 1957; MacDonald 1982; van Deman 1912) and still occupy a considerable number of scholars. Very little, however, is known about the distribution, development and use of ‘pre-Roman’ fired bricks in Europe, i.e. bricks pre-dating the significant breakthrough that took place in early Imperial Rome. Henrik Gerding (2006) has written a preliminary account, and an in-depth study is forthcoming (Gerding and Östborn forthcoming).

There are several intriguing features of the limited use of fired bricks in Hellenistic Europe. Clearly, the possibility of using fired bricks was known from literary sources, as described above. Also, the use of roof tiles was common already in the Archaic period. Why did fired bricks not appear until the Hellenistic period? Soon after their first appearance, fired bricks were found all around Hellenistic Europe. Nevertheless, their use was seldom continuous at a given place. Fired bricks seem to come and go. Why? As bricks spread around the region, they quickly evolved into a variety of forms (Fig. 1). Purpose-made bricks with a wide variety of sizes and shapes were used in walls, columns and arches. They were used to build tombs, city walls and baths. What does this diversity tell us about the evolution process?
Fig. 1

Examples of Hellenistic fired bricks. a Brick wall of the so-called Naumachia at Tauromenion, Sicily (possibly Augustan). b Brick column of the Basilica at Pompeii (second half of the second century BCE). c Vaulted brick tomb at Rhegion, Calabria (third or second century BCE). d Brick pavement in Casa del deposito a volta at Soloeis, Sicily (after ca 250 BCE)

The sporadic appearances of fired bricks make the hypothesis conceivable that they were ‘invented’ independently many times and used at random locations and in random ways. Keeping this null-hypothesis in mind, the spread of fired bricks across Hellenistic Europe is ideally suited to study with general similarity networks, given the seemingly complex evolution of the process over an extended period of time, its spatial nature and the considerable number of attributes of different kinds that have to be used to characterise each brick context.

Finally, to give a feeling for the subject, it should be noted that Hellenistic fired bricks differ considerably from their modern equivalent in both size and weight. An average post-medieval brick (25 × 12 × 6 cm) weighs only about 3.1–3.6 kg whereas a typical Hellenistic fired brick measuring 35 × 35 × 10 cm (a large tetradoron) weighs about 20.1–24.5 kg. These bricks could be managed only with both hands whereas modern bricks can be held in one hand, leaving the other one free to operate a trowel.

Diffusion of Innovations

The spread of fired bricks across the Mediterranean is an example of the diffusion of innovations. This is an active field of research in several academic disciplines. The standard reference is Everett Rogers’ monograph (2003), first published in 1962. Carlo Cipolla (1972) was among the first to apply the perspective to the more distant past, considering technical innovations in seventeenth-century Europe. He subscribed to the general view that social contact is essential—the mere observation of the innovation, or a blueprint thereof, is usually not enough for adoption.

The cultural geographer Torsten Hägerstrand stressed the importance of the spatial aspect of the diffusion process, supporting his view with empirical data from Swedish agriculture. He considered individual links between potential adopters, whose spatial locations corresponded to the farms where they lived. He employed stochastic algorithms to model the process in which these links formed a spatial network through which the innovation spread. Hägerstrand was a pioneer, but his influence on the field was not immediate, since the translation into English of his doctoral dissertation from 1953 was delayed (Hägerstrand 1967).

Hägerstrand’s perspective is very much in line with our own. We also consider a spatial network (where the localised brick contexts take the part of the farms). There are two differences though. First, we do not attempt to reconstruct the underlying social network in which the innovation spread, only the diffusion network formed by the links in such a network along which the spread was successful. Second, in the present study, we do not address the question of the nature of the contact that led to the spread of fired bricks from one site to another. A single person-to-person contact may sometimes have been crucial, but many people were probably involved in most cases. In network studies such as the present, where a node may correspond to an entire town, the term societal network may be more appropriate than a social network.

The diffusion network is a subset of the societal network of contacts between sites (Fig. 2). Two sites are linked in the societal network whenever there is a contact that potentially may lead to the transfer of the innovation from one to the other. These contacts may appear and disappear, so that the societal network is time-dependent. The set of edges of the diffusion network is the union of the successful transfers from one site to another, and its nodes are the sites at which the innovation sometimes was used. Several sites may have contributed to the transfer to a single new site, so that several edges may correspond to a single transfer. If the only source of information is the material record of the innovation, all we can learn about the societal network is gleaned indirectly from the diffusion network, which in turn has to be approximated by similarity networks.
Fig. 2

The diffusion network (grey edges) of successful transfers of the innovation as a subset of the social or societal network (black edges) of contacts between sites

A different way to model the diffusion of innovations spatially is to use continuum models governed by partial differential equations, in much the same way as diffusion processes are modelled in physics or chemistry (Kandler and Laland 2009; Kandler and Steele 2009). Such models correspond to the diffusion of innovations in social networks that resemble lattices (Fig. 3, panel a), where the innovation spreads from neighbour to neighbour like circles on the water. They exclude by construction the possible influence of more complex social network structure.
Fig. 3

Four kinds of network structure. a Lattice. b Random network. c Small-world network. d Scale-free network. All four networks contain the same number of nodes and edges. The edges are just distributed differently

Current research about the diffusion of innovations deals with the psychological and sociological processes that lead from the knowledge of an innovation to the decision to adopt or reject it (Rogers 2003), as well as factors that determine whether the innovation is adopted or rejected at a larger societal scale (Wejnert 2002). The role of the social network structure for the success or failure of the diffusion process is also investigated (Watts 2002). The first treatise of the subject with network focus was that of Valente (1995).

In this study, we do not touch upon these deeper aspects of the field, but rather treat the spread of fired bricks as a matter of fact and analyse its characteristics. We return to the subject in a forthcoming monograph (Gerding and Östborn forthcoming) and in a paper where we try to fit societal network models to the archaeological data (Östborn and Gerding forthcoming).

Network Characterisation

In the analysis presented in this paper, we refer repeatedly to a number of quantities and concepts that characterise networks and the nodes within them. These quantities and concepts are defined in the glossary provided in the introduction to this special issue (Collar et al.2015). Even more information can be found, for example, in Albert and Barabàsi (2002), de Nooy et al. (2005) and Valente (1995).

Here we discuss briefly how to characterise the overall structure of a network, since this structure is central when we try to infer characteristics of the diffusion process as a whole, and the social network in which it took place. There are four basic network types.

Lattices

Few real networks look like a regular lattice (Fig. 3a), but it may be used to model social networks in a geographical region where people have contact only with their nearest neighbours. The clustering coefficient is typically large, and the average shortest path length is also large—it takes many steps to reach from one end of the lattice to the other.

Random networks are the opposite of lattices. They represent no spatial structure whatsoever, and the probability that two nodes are connected is the same for all node pairs (Fig. 3b). The clustering coefficient is often small (but depends of course on the overall number of edges), and the average shortest path length is also small.

Small-world networks are characterised by a large clustering coefficient and a small average shortest path length. If we consider a lattice and replace a small fraction of the edges by random, possibly long-range edges, the average shortest path length decreases drastically, whereas the clustering coefficient remains large. This is the classical model of a small-world network (Fig. 3c).

Scale-free networks are characterised by the existence of hubs with much higher degree than would be expected by pure chance (Fig. 3d). The degree distribution f (D) gives the number of nodes in the network that have degree D. In a scale-free network, the degree distribution is a power law f (D)∼D-a, where a > 1 is a constant. Such degree distributions arise only if there is some mechanism that favours large degrees. One popular model is that of an evolving network where new edges have a larger chance to be attached to nodes that already have a high degree. This is called preferential attachment. In contrast, if new edges are attached randomly and do not care about degree, the degree distribution will be approximately normal for fairly large degrees: \( f(D)\sim \exp \left(-\frac{D^2}{\sigma^2}\right) \), where σ > 0 is the standard deviation.

Nothing prevents a small-world network from being scale-free, and vice versa. However, scale-free networks sometimes have a rather small clustering coefficient, so that it is not a small world, and small-world networks often have an approximately normal degree distribution, so that they are not scale-free. Thus, to characterise an empirical network in these terms, it should be clearly stated whether it is a small world, scale-free, or both.

As an example, a social network with both short- and long-range connections that has a rather democratic structure—where no person, or group of persons, is much more important than any other—is a small world but not scale free. In a hierarchical social network where influence radiates from one or more hubs, the structure tends to be scale-free without being a small world.

Material Record and Methods

The Database

An extensive literature search supplemented with personal observations yielded a catalogue of 275 Hellenistic fired brick contexts at 131 sites. Out of the 275 contexts listed in the catalogue, only 233 contexts at 113 sites were treated as reliable material for network analyses and other kinds of analyses. The catalogue will be published shortly (Gerding and Östborn forthcoming).

All brick contexts from the period 600–1 BCE in the Mediterranean area or the period 350–1 BCE in the Near East were accepted in the catalogue. The vast majority of the contexts found are located in the region we call Hellenistic Europe (Fig. 4) and belong to the period 350–1 BCE. Most contexts are located in present-day Italy and Greece, whereas some fewer contexts are found in Albania, Bulgaria, Turkey and France (ordered according to decreasing abundance).
Fig. 4

Sites with reliable fired brick contexts in Hellenistic Europe. The non-European sites in the catalogue are: Alexandria and Karnak in Egypt, Nippur and Seleucia on the Tigris in Iraq, and Aï Khanoum in Afghanistan (not shown)

Outside Europe, we found brick contexts from the relevant period at two sites in Egypt, two sites in Iraq and one site in Afghanistan. We are more confident that our search for fired bricks in Europe has yielded a representative sample than the search outside Europe. Therefore, we restrict our conclusions and hypotheses to Europe, although all reliable contexts are included in the analysis (Fig. 5).
Fig. 5

Temporal distribution of fired bricks. The phases are 25 years wide and are centred at the indicated years. Out of 233 reliable contexts, 54 are too poorly dated to be assigned a phase and are not included. The relative abundance of the three structural uses of fired bricks is shown. Bricks used for masonry are the primary target of this study, but it cannot be excluded that bricks employed for casing and pavement are part of the same diffusion process. Compare Figs. 7 and 13

In Table 1, we present the attributes that describe each of the brick contexts. We also classify each attribute according to Table 1 in Östborn and Gerding (2014).
Table 1

The attributes in the database, their type and their possible values. Out of 18 attributes, 13 describe the properties of the brick constructions per se, whereas two describe its temporal location and two describe its spatial location

Attribute

Attribute type

Possible values

Reliable

Categories

Yes/No

Site

Categories

One of 131 ancient site names

Location

Numerical vector

Any pair of real numbers that encodes latitude and longitude

Dating

Interval

Any interval of years contained in the interval 600–1 BCE

Phase

Numerical value

600, 575, 550, …, 50, 25, 0 (BCE)

Structural use

Categories

Casing/Masonry/Pavement

Context

Categories

Domestic/Manufacture/Military/Public/Sacred/Sepulchral

Function

Categories

Arch/Basin/Barrel vault/Bench/Bonding course/Cist/Column/Courtyard/Cover/Door frame/ Floor/Foundation/Half-column/Hypocaust/Inner wall/Niche cover/Pillar/Pillar base/Quoin/Street/Terrace wall/Underground wall/Upper wall/Wall/Wall socle/Water channel/Furnace wall

Structure

Categories

Brick-faced concrete/Imbrices and mud/Interlaced brickwork/One-stone brickwork/Mixed materials/Solid brickwork

Binding

Categories

Clay mortar/Mortar/No mortar/Timber bindings

Shape

Categories

Circle sectors/Circular/Curved/Grooved/Imbrex/Pierced circular/Rectangular/Semicircular/Square/Tegula/Triangular/Voussoir

Size category

Categories

Lydion/Pentadoron/Sesquipedalis/Retradoron

Thickness

Interval

An interval given with half-centimetre precision

Plaster

Incidence

Yes

Stamp

Incidence

Yes

Poorly fired

Incidence

Yes/No

Tile bricks

Incidence

Yes/no

Combined with mud bricks

Incidence

Yes/No

Apart from the values listed above, all attributes except the four first ones may also have the value Unknown (or Undefined). That the value of a given attribute is unknown in two contexts is not treated as a similarity and cannot contribute to fulfil a condition for connection. Therefore, Unknown is not a value on the same footing as the other ones.

The exactness of the dating varies considerably among the contexts. The width of the interval defining its value varies from 3 years to over 300 years. We define another temporal attribute called phase. It is the best estimate of the period of construction, given as bins that are 25 years wide. For instance, phase 150 means that the brick context was more probably constructed between 163 and 137 BCE than in any other bin. For some poorly dated contexts, it is impossible to estimate the phase. Then the value Unknown is assigned for this attribute.

The dating is negatively defined in the sense that it excludes time periods in which we know the brick context was not constructed. In contrast, the phase is positively defined in the sense that it is the best estimate of the period of construction. Consequently, the two temporal attributes play different roles in the construction of similarity networks. Overlapping dating can be seen as a necessary condition for a direct causal relationship between two contexts. This similarity criterion is therefore almost always used as part of the overall condition to connect two contexts. In itself, the criterion gives rise to many false positives. The less exact the dating of a context, the more connections are possible to other contexts. To avoid such excess connections, we can add the criterion that the phase difference should be less than some given value. The use of the phase in the similarity condition excludes all very poorly dated contexts. The resulting networks may therefore be used as reference when spatio-temporal processes are inferred. The phase can also be used to create time slices, for instance, similarity networks that are divided century by century.

The assignment of size categories is an attempt to fit the bricks into commonly used standard dimensions in the Graeco-Roman world. The Lydion measured one foot by one-and-a-half. The Pentadoron was five palms on each side, the Tetradoron four palms. The term Sesquipedalis is used in the present study to denote square bricks one-and-a-half-foot wide. Since the length of a palm and a foot varied, we employed intervals of tolerance. For instance, any brick measuring (44−53) × (28−35.5) cm was assigned the value Lydion to the attribute Size category, provided the length-to-width ratio did not deviate from 3:2 by more than 2 cm. The Thickness attribute gives the thinnest and thickest bricks occurring in the context as an interval, in which the uncertainty in the measurement is included.

The incidence attributes Plaster and Stamp are binary in essence but have only one possible value Yes, apart from Unknown. We can never be certain that a brick construction was not covered with plaster, since it may have been worn off. Stamps may also have been worn off, or we might not have found those bricks in a building that were actually stamped. That Combined with mud bricks has the value Yes means that fired bricks were used together with mud bricks. Similarly, that Tile bricks has the value Yes means that the bricks were made of reused roof tiles.

The database has several hierarchical elements, meaning that the value of one attribute limits the possible values of another. The structural use of a context determines the function to a considerable degree. For instance, bricks for which the value of Structural use is Pavement cannot have the value Wall of the attribute Function. The function in turn determines the shape of the bricks, to a large extent. For instance, the value Wall of the attribute Function excludes the value Circular of the attribute Shape. Furthermore, only rectangular or square bricks can be assigned a size category. Even more, the value Rectangular of the attribute Shape excludes the values Pentadoron, Sesquipedalis and Tetradoron of the attribute Size category, just as the value Square excludes Lydion.

Apart from the hierarchies, there is another complication in the database. The attributes Function, Shape and Size category can have more than one value in each context. For example, we may want to keep a building together as a single context, but this building may contain bricks with different functions and shapes. In Apollonia, there is a brick cistern from 300 to 200 BCE where fired bricks are used to construct a Basin and a Water channel. They belong to the size categories Pentadoron and Sesquipedalis. Sometimes it is hard to discriminate brick contexts at the same site. Should a group of similar tombs from the same date be seen as one context or many? We tried to use common sense, but it is impossible to give a clear-cut criterion how to make the decision.

About the Statistical Analysis

All employed statistical tests were random permutation tests like those outlined in Östborn and Gerding (2014). We also calculated several statistical distributions (degree distributions and edge length distributions). No statistical tests were applied to these, since the features we were interested in were clear in visual inspection, sometimes facilitated by the introduction of logarithmic scales.

The use of statistics in archaeology may give a false air of exactness, since it is hard to know whether the data at hand is a representative sample. Correlations that biased our final selection of data about Hellenistic fired bricks may have occurred at several levels. First, there may have been correlations between the sites archaeologists have chosen to excavate or investigate. Second, there may have been correlations between the finds they chose to report. Fired bricks have often been peripheral in archaeological field work. A more or less random report of a few bricks might have inspired the report of similar fired bricks at a nearby site. Third, there may be correlations between the reports of fired bricks that we happened to find in the literature. The mere following of a reference from one report of fired bricks to another creates a correlation.

The severity of the last problem is diminished by the fact that the literature search was extensive. It went on for several years and approached the literary corpus from several angles. These included careful search of general literature treating Hellenistic architecture, construction technique or building materials. It also included systematic word-based searches in archaeological databases, as well as continuous follow-ups based, for example, on the perceived likelihood of finding reports on fired brick at certain sites or in certain kinds of contexts or publications.

Network Analysis Software

We did not find software that fulfilled the perceived needs for this study. Neither geographical information systems nor social network analysis programs such as Pajek (de Nooy et al.2005) provide the versatile commands needed to create and analyse general similarity networks from a database with a complex set of attributes, such as ours (Östborn and Gerding 2014). Therefore, we wrote our own network analysis program in the numerical computing environment MATLAB. The software was custom-made for our database and those questions we were interested in. Since it cannot be used by other archaeologists in its present form, we do not describe it here.1 The routines for the statistical tests were also written in MATLAB, and various numerical manipulations of the network output were also performed in that environment.

Results

Exploratory Network Analysis

Figure 6 shows the evolution of the distribution of fired bricks. It is not easy to recognise any spatial patterns apart from the origin in the North Aegean in the fourth century BCE and the growing number of brick contexts that are found in northern Italy in the last decades BCE, a precursor to the explosion of fired bricks in Imperial Rome. It is clear that the spread did not occur gradually as a wave, like the spread of agriculture across Europe. If there was a causal diffusion process, it quickly travelled large distances, jumping back and forth.
Fig. 6

Evolution of the spatial distribution of fired bricks. Only those contexts that are well enough dated to be assigned a phase are shown. The phases are 25 years wide and are centred at the indicated years. The contexts in Aï Khanoum, Herakleion and Seulecia on the Tigris are assigned the phase 300 BCE, the contexts in Nippur 275 BCE and the contexts in Karnak 225 BCE (not shown). Compare Fig. 5

A relatively large number of contexts are found in Calabria and Sicily, but these are distributed across the centuries; fired bricks do not seem to have been a dominant building material in this region at any time, and they were seldom continuously used at a given site.

Choice of Networks

To be able to discern more structure in the process, we studied similarity networks defined by the criterion that a pair of contexts are connected if and only if their dating overlaps, and they have at least X attribute values in common, where X was varied. This can be seen as the default family of networks in the analysis presented below.

That a pair of contexts has an attribute value in common means that their values overlap when the attribute is of the interval type (Table 1). If the attribute is a numerical value, a numerical vector, a category or an incidence, it means that the values are the same. That two contexts have identical location (are found at the same site) is not treated as a similarity in the analysis.

Naturally, the number of edges decreases as we let X increase (Fig. 7). For small enough X, all contexts are connected to all the others, and for large enough X, all contexts are isolated. Consequently, there has to be an intermediate, critical level Xc of similarity above which the network fragments into smaller components. In our database, this happens at Xc = 9.
Fig. 7

The family of networks defined by the condition that two contexts are connected if and only if their dating overlaps, and they have at least X attributes in common, no matter which (Table 1). Contexts from all times and all regions are connected in a large component for X ≤ 9. The network for X = 9 is called the critical network in the family. The second largest network component is shown in blue. Isolated contexts not connected to any other are shown as hollow circles

As X increases from zero, the average shortest path length and the diameter of the network first increase, since it becomes harder to find a short path between two nodes when edges are removed. Above the critical level Xc, the network is fragmented into components that get smaller as X increases further. No long paths can then be found, and the average shortest path length and the diameter decrease again. In other words, the topological size of the network peaks at Xc. For the sequence X = {5, 6,7,8,9,10,11,12,13} the diameters are D = {5, 6,7,8,11,8,4,4,2}. The last six of these networks are shown in Fig. 7.

The aim was to mimic the diffusion network (Fig. 2) by similarity networks. The diffusion of fired bricks went on for over 300 years before making a breakthrough, covering many generations of brick producers, architects and decision-makers. The number of social steps leading from the first adoption to the last must have been large. Similarity networks with large diameters must therefore be the most appropriate ones. Assuming that there was indeed a single diffusion process leading from the North Aegean in the fourth century BCE to the city of Rome at the beginning of the Common Era, the best choice is therefore to focus on the critical similarity level, in our case X = 9. This choice is also motivated by the fact that we should minimise the number of ‘false-positives,’ the number of edges that do not correspond to a close causal link. Therefore, we should focus on the highest possible level of similarity.

This argument also motivates our choice of default family of networks. Overlapping dating is necessary for the social contact that defines a diffusion process, but, apart from that, we should introduce as few a priori restrictions as possible. Say, for instance, that we judge that only fired bricks used for masonry are important in the diffusion process, so that we require that all contexts in the network have the value Masonry of the attribute Structural use (Table 1). Or, say that we judge the attribute Stamp to be irrelevant, so that it is excluded from the calculation of the similarity level of two contexts. Such supplementary conditions for connection means that we have to lower the general similarity level X to prevent fragmentation and thus accept a larger number of false edges. Nevertheless, we tried a variety of network families at the edge of fragmentation, according to the philosophy that only network properties robust to variations deserve to be highlighted and interpreted.

The largest diameter, D = 11, obtained at the critical similarity level X = 9, most certainly represents fewer steps than the number of direct causal contacts that lead from one end of the real diffusion network to the other (Fig. 7). The obvious reason for this is the incompleteness of the database. Many brick contexts are unknown to us, and many of the known contexts are poorly documented. To bridge these missing data and still get a non-fragmented network, we have to lower the similarity level X, just as we have to do if we introduce supplementary conditions for connection. This operation reduces the diameter. Many contexts that become connected by a single edge in the similarity networks were, in reality, causally linked only via intermediate contexts that are either unknown to us, or too poorly documented to be part of a large network component. Thus, even the best similarity network cannot be trusted in its details.

Network Interpretation

In non-fragmented similarity networks, there are some small components, apart from the dominant one that connects the bulk of the contexts. There are also a number of isolated contexts that are too dissimilar, or too poorly documented, to be connected to any other context. In Fig. 7, the largest of the small components are shown in blue for X = 8 and X = 9.

Some of these can be interpreted neatly. For X = 8, the blue component consists of contexts where bricks made of roof tiles have been used to construct wall socles supporting mud bricks. These contexts are connected to a strand stretching from south to north, connecting the sites Lokroi Epizephyrioi, Kaulonia, Fregellae and Suasa. The dating of these contexts is consistent with a diffusion process in which this particular way to employ tile bricks spreads northwards across Italy during the third and second centuries BCE. In effect, the similarity network has yielded a seriation, even if the amount of data is too small to allow any definitive conclusions.

For X = 8, the blue component in Fig. 7 represents another possible micro-diffusion, in which the use of columns made of circular bricks spread from Calabria westwards along the northern shore of Sicily during the same time period, from Kaulonia via Messana and Tyndaris to Halaesa, before it finally crossed the Tyrrhenian Sea northwards to reach Pompeii.

Let us turn to the overall diffusion process. We cannot expect a single serial process; the evolution is complex and branches into many different forms, which sometimes recombine (Östborn and Gerding 2014; Fig. 5). One thing we can do, however, is to look for the shortest paths in the networks that lead from the first fired bricks in the North Aegean to their final arrival in Rome. Such paths in the diffusion network can be interpreted as the main channels of flow in the diffusion, and its properties reveal something essential about the process. In the similarity networks, we have to look for robust properties of these paths and hope that these match properties of the main channels of flow in the diffusion network.

Figure 8 shows such paths in the critical default network (Fig. 7). In this network, the earliest context that is part of the main component is a tumulus tomb in Seuthopolis with inner walls made of fired bricks. According to our best information, it belongs to the phase 350 BCE. One of the latest contexts in our database is found in Rome. It is dated between 18 and 12 BCE, and it is also a tomb with inner walls made of fired bricks. The shortest paths between these two contexts are shown. Very similar paths are obtained if another context in Rome is chosen. (All contexts in Rome belong to the phase 25 BCE.)
Fig. 8

The shortest paths between the earliest and latest contexts in the critical network shown in Fig. 7 (X = 9). Following the temporal ordering, the path starts in Seuthopolis, passes Rhegion in Calabria, Morgantina on Sicily, jumps north to Aquileia close to Trieste, before reaching Dyrrachium in Albania. From here, there are several paths. One group goes to one of a set of sites in Northern Italy before it turns south to the region around Rome, the other jumps back to Sicily (Kentoripa) before it reaches Rome. All these shortest paths contain eight nodes connected by seven edges. Those sites that are part of at least one shortest path are coloured black

Naturally, if another critical network is chosen, the paths change. We tried, for example, the critical network in which it is demanded that all contexts are well enough dated to be assigned a phase, the critical network where we used no temporal condition whatsoever, and networks where only bricks used for masonry were included. There are some robust properties of the paths in these networks that deserve to be highlighted as possible reflections of the true diffusion process. First, the diffusion does not seem to be spatially gradual; the paths jump back and forth in the region. Second, from the first contexts of fired bricks in the North Aegean, the main diffusion paths go to southernmost Italy, often to Rhegion in Calabria or Morgantina on Sicily, sometimes passing Apollonia in Albania on their way there. Third, most, but not all, paths arrive in northern Italy in the second century BCE before taking the last step south to the region around Rome. Fourth, Dyrrachium in Albania is a common site passed by the paths on their way between southern and northern Italy in the second century BCE. Fifth, Greece is seldom part of the shortest paths; it seems peripheral to the main flow of diffusion. (However, baths with fired bricks from the third century BCE in Gortys or Oiniadai in western Greece are sometimes parts of the shortest paths alongside the contemporary bath in Morgantina, Sicily.)

Apart from the geographical routes taken by the diffusion, the shortest paths tell us something about the main evolution of the properties of fired bricks and the way they were used. Along these paths, almost all bricks are rectangular with mortar used as binding. Most are used in walls. The thickness of these bricks seems to decrease gradually, from around 10 cm to below 5 cm. In the early days, the construction technique was mostly one-stone-brickwork, whereas brick-faced concrete dominated later on. These findings are not surprising per se, since the basic way to use bricks is to construct walls with rectangular bricks. The trends whereby bricks become thinner and that brick-face concrete starts to dominate in Roman times can be readily seen in the material without the use of network analysis. What is interesting, though, is that these basic properties and trends completely dominate the shortest paths. The more unusual and specialised ways to use fired bricks might therefore be seen as thinner branches that emerge from this main stem of evolution more or less independently at different times and at different locations.

Early and Late Phases of Brick Usage

The networks in the default family fragment into two components of comparable size at similarity level X = 10 (Fig. 7). One is centred in the southeast and the other in the northwest. Interestingly, the two components are also temporally separated. In the southeast component, no context is assigned a phase later than 200 BCE, and in the northwest component, no context is assigned a phase earlier than 75 BCE. Among the poorly dated contexts, however, there is one in the early southeast component with a late dating of 150–1 BCE, and one in the late northwest component with a rather early dating 148–1 BCE.

The fragmentation of the network into one early and one late component may have a trivial explanation, since there are few contexts from the middle of the Hellenistic period (Fig. 5). Only three contexts are assigned the phase 175 BCE. The network condition of temporal neighbourhood of connected contexts makes it hard to bridge this bottleneck with edges.

However, the same kind of fragmentation appears even if we explicitly remove all temporal conditions (Fig. 9). Two components appear in network B, defined by the condition that a pair of contexts is connected if and only if they have at least nine attributes in common among those that are neither Dating nor Phase. The spatial characteristics of the components are the same as in the default network A. More interestingly, the temporal characteristics also stay more or less the same, even if there are a few late contexts in the early southwest component and a few early components in the late northwest component.
Fig. 9

For similarity levels X just above the critical level Xc, the dominant network component splits into two components of comparable size, one component with early contexts in the southeast and one component with late contexts in the northwest. This is a robust feature that appears in all reasonable networks. Network A: X = 10. The dating of two connected contexts must overlap. Network B: X = 9. The dating and the phase are not considered when the similarity level is calculated. Network C: X = 8. The phases of two connected contexts must be defined, and the phase difference between them must not be greater than 50 years. Their dating must also overlap

This means that the two components must correspond to qualitatively different ways to use bricks, apart from their spatial and temporal separation. Since our network conditions typically mix different attributes, it is not immediately possible to pinpoint the difference. The clearest line of separation is found in the attribute Context (Table 1). In the early component, more than 75 % of the contexts are Sepulchral, regardless the details of the network definition. In the late component, more than 75 % of the contexts are Military or Public. As we let the required similarity level X increase further, this trend is strengthened. All contexts in the early component become sepulchral, and all contexts in the late component become military or public, with military contexts dominating.

This does not mean that military or public use of fired bricks did not occur in the early Hellenistic phase, or that bricks were not used in tombs in the late Hellenistic phase. It means that the early phase was characterised by a standardised way to use fired bricks in tombs, and that the late phase was characterised by a standardised way to use fired bricks in military or public buildings. Military and public buildings typically have a larger physical scale than sepulchral buildings and are generally commissioned at a higher societal level. A possible interpretation is that the use of fired bricks became established and standardised higher up in the societal hierarchy in the late Hellenistic phase than it was in the early phase.

Not only was the field of application of standardisation different, the degree of standardisation seems to have been greater in the late Hellenistic phase than in the early phase. The late component in northern Italy survives longer as the similarity level X increases than does the early component (Fig. 7). More generally, late components are typically characterised by a larger clustering coefficient than early components, which reflects the fact that the member contexts in late components have more homogeneous attributes. (In a component of perfectly homogeneous contexts, all contexts are connected to all the others, leading to a clustering coefficient that equals 1.)

It is not possible to decide from the empirical data whether the existence of two robust components means that there were two independent, sequential diffusions of fired bricks, or that there was a single process that was close to dying out shortly after 200 BCE, but slowly recovered. In any case, it may be interesting to look for potential bridges between the early and the late stage of brick usage.

In Fig. 8, Aquileia and Dyrrachium link southern and northern Italy along the shortest paths from Seuthopolis to Rome. These military contexts were probably not constructed earlier than mid-second century BCE and are thus appropriately dated to be causal connections between the southeast early component and the northwest late component.

Inspecting the late contexts in the early components in Fig. 9, we see some contexts in northern Italy that may act like seeds for the late component. In networks A and B, we find an underground vaulted chamber tomb in Sarsina, dated 230–80 BCE, similar to those found further south in Rhegion. In network C, we find two city walls in Arretium and Ravenna. These belong to the third century BCE and may seem too early to be seeds for the late component. However, there may be unknown contexts that bridge the temporal gap between these and the late component. In all three networks, a vaulted brick tomb in Apollonia, Albania, from the phase 200 BCE, appears as a late context in the early component. This strengthens the idea, emerging from Fig. 8, that present-day Albania played an important role in the diffusion between south and north. The nature of this role is not entirely clear, however, as will be discussed below.

Further south, in Tyndaris, Sicily, there is a domestic context in the early component dated no earlier than 150 BCE. Very tentatively, the northern shore of Sicily may be seen as an ‘exit door’ for the hypothetical diffusion northwards. This picture is supported by the abovementioned hypothetical micro-diffusions towards the north from the southernmost part of Italy of brick columns, and of wall socles made of tile bricks.

Central Sites

Another way to find gateways between the early and late phase of Hellenistic brick use is to identify central nodes in critical networks where there is still one dominant component. Loosely speaking, these should be in the middle of the diffusion process. We employ three measures that quantify the centrality of a node: closeness centrality, betweenness centrality and degree (Collar et al.2015).

We should distinguish between closeness and betweenness centrality on the one hand, and degree on the other. Degree is a local measure that describes the relation between the context and its immediate topological neighbourhood, similarity-wise. The other two measures are global in the sense that they describe the relation between the context and the rest of the network. Since we are interested in the diffusion process as a whole, these measures are more appropriate.

We do not attempt to decide from a theoretical point of view whether closeness or betweenness centrality is the better measure to identify key contexts in the diffusion. It turns out, however, that there are most often one, two or three contexts that have much higher betweenness centrality than the rest, whereas many contexts have a similar, high closeness centrality. Thus, betweenness centrality is the measure that is better suited to tell important and unimportant contexts apart. Contexts with high betweenness centrality can be interpreted as bottlenecks. If they were hypothetically removed, or had contra-factually refused to adopt fired bricks, the diffusion could have stopped or evolved quite differently.

Figure 10 shows central contexts in two critical networks, where the early and late parts of the diffusion are still connected into a single component. Figure 10a represents the same network as that in Fig. 8, where the shortest paths from the beginning to the end of the process are shown. The three contexts in Morgantina, Aquileia and Dyrrachium that are part of these paths are also the contexts with the highest closeness and betweenness centrality. The betweenness centrality of these contexts is significantly larger than of all other contexts.
Fig. 10

Central contexts in two critical networks (for which X = Xc). The three contexts with largest betweenness centrality (opposing arrows) and closeness centrality (centred circle) are shown, as well as the hub with the largest degree (star). aX = 9. The dating of two connected contexts must overlap. bX = 7. The phases of two connected contexts must be defined, and the phase difference between them must not be greater than 50 years. Their dating must also overlap

The central context in Morgantina (walls and door frames in domestic houses) is rather poorly dated. Such a context may become central for the trivial reason that the dating overlaps the dating of many other contexts, to which it potentially may be connected. To avoid this effect, the network represented in Fig. 10b was investigated, where we required that if two contexts were to be connected, the phases of both must be defined, and the phase difference must not be too large. (Network C in Fig. 9 belongs to the same family.) The price to pay is that we have to lower X to get a non-fragmented network. It is a trade-off between better temporal accuracy and worse accuracy in the similarity of the other attributes.

In this network, the central contexts are different from those in the network represented in Fig. 10a, but they are still placed in southernmost or in northernmost Italy, where those in the south tend to be dated before those in the north, supporting the view that the diffusion jumped northwards in the second century BCE.

Furthermore, there is no context in Dyrrachium that is a significant betweenness centre, but there is still one context with large closeness centrality. The role of Dyrrachium in the hypothetical jump to the north in the second century BCE remains unclear. In Fig. 8, the shortest path first jumps all the way up from Sicily to Aquileia, then back south to Dyrrachium. The central contexts in Dyrrachium shown in Fig. 10 are later than those in the north, contradicting the idea that the diffusion to the north passed Dyrrachium on its way. Dyrrachium is one of few sites where the use of fired bricks seems to have been continuous throughout the Hellenistic period. Sites in Albania may therefore have acted like continuous seeds for diffusion in every direction, albeit weak, rather than a region through which the diffusion passed in a given direction.

Are the identified centres really bridges between the early and late components? In the network represented in Fig. 10a, the match is perfect: The central context in Morgantina belongs to the early component in the corresponding network A in Fig. 9; the context in Dyrrachium belongs to the late component, whereas the context in Aquileia belongs to neither, acting like the bridge over which the shortest paths shown in Fig. 8 pass from the early to the late Hellenistic phase.

In the network represented in Fig. 10b, the match is not as good. The betweenness centre in Kaulonia does belong to the early component in the corresponding network C in Fig. 9, but the central contexts Placentia, Regium Lepidi and Dyrrachium belong neither to the early, nor to the late component. They are either too dissimilar or too poorly documented to show up.

Ariminum is the main hub in both networks represented in Fig. 10. This site also has the highest degree in many other critical networks and often becomes the main hub in the late, northwestern component of fragmented networks. This is satisfactory, since Ariminum had a crucial position in the communication system of Roman Italy. From here, Via Flaminia led to Rome and Via Aemilia to Placentia and other major sites in the Po valley. The local character of the degree measure becomes evident in the case of Ariminum, since the brick contexts in Ariminum are late and cannot be central to the diffusion process as a whole.

Among the early contexts, there is no single site that dominates in terms of degree as much as Ariminum does among the late contexts. However, one of the contexts in Rhegion often becomes the major hub. This situation is indicated in the early components of networks A and B in Fig. 9, where edges connect Rhegion and Messana to other sites like the rays of a star. That these sites appear as local centres is satisfactory in the same way as in the case of Ariminum: The Strait of Messina was an important passageway.

Statistical Analysis

Diffusion or No Diffusion?

In the broadest sense of the term, the features of a set of contexts is the outcome of diffusion if and only if there are causal links between the contexts such that the features of each is linked to the features of every other by a chain of cause and effect. If there is no such set of linked contexts, the features are random or have a common, background cause. Similarity network analysis is meaningful only if we can exclude this null hypothesis, or, at least, demonstrate that it is very unlikely.

To that end, we used spatial random permutation tests to see whether similar contexts are closer than expected by chance (Östborn and Gerding 2014; Fig. 6). If the difference is significant, the null hypothesis can be rejected. The median edge lengths in the default family of networks (Fig. 7) were compared with the mean median edge lengths in 5,000 randomised networks defined by the same network condition as the true network, but where all contexts at each site changed location collectively to another site in a random permutation.

The median edge length was used in the analysis rather than the mean because it downplays the importance of faraway outliers, such as the site Aï Khanoum in Afghanistan. Due to the incompleteness of the data, we wanted to make the quantities employed in the statistical analysis as little dependent on individual finds as possible. The median is more robust than the mean in this sense. Only non-zero edge lengths were included in the calculation of the median, since it was the distance between similar contexts at different sites that was the focus of our interest.

All contexts at a given site were kept together in the spatial randomisation since it is sometimes hard to discriminate contexts at the same site, as discussed in the database section. This means that two contexts at the same site sometimes have a closer relationship than two contexts at different sites. If they had been moved independently to new random sites, the broken close relationship between them would have biased the analysis.

Figure 11 shows that the median edge lengths in the true networks are much smaller than the mean median edge lengths in the randomized networks. The difference is highly significant (p = 0.00005 for X = 9). Moreover, the more similar the contexts, the closer they are. This is not the case when the locations are random, of course. (The upshot of the curve for X = 13 is the result of poor statistics, which in turn is due to the fact that very few context pairs are similar enough at this level of X to be connected by edges.)
Fig. 11

Median edge lengths in the default family of networks (Fig. 7) with similarity level X (solid curve with circles). Also shown are the corresponding average median edge lengths in networks where the locations are randomly permuted (solid curve), as well as the median edge lengths that correspond to deviation from randomness at significance level p = 0.05 (dashed curves). The 5,000 random permutations were performed for each X to compute average median edge lengths

It can be argued that the tendency seen in Fig. 11—that the median edge length decreases when the similarity level X increases—is a necessary and sufficient condition for diffusion, assuming that the diffusion is preferably short range. There are two reasons for this fact. First, the number of edges that do not correspond to a causal link is expected to decrease as X increases. These false-positives have no preferred edge length. Therefore, the proportion of true, preferably short, edges increases, so that the median edge length decreases. Second, the attributes can be expected to change gradually in a causal chain of adoptions of fired bricks. Therefore, in a causal chain of adoptions A → B → C at sites A, B and C, the proportion of edges that correspond to a second-hand link A–C decreases as X increases. Such edges are longer in the mean than edges corresponding to first-hand links A–B or B–C. Therefore, the median edge length decreases.

Confounding Covariates

The above analysis does not exclude the possibility that the deviation from randomness results from common causes that make nearby contexts similar even if there is no causal link between them. This is the problem of confounding covariates (Östborn and Gerding 2014). In such a situation, there are regional differences in the building tradition in general that influence the way fired bricks are used.

The attribute in our database (Table 1) that most clearly reflects such regional traditions is Size category. These standard dimensions were used for mud bricks also. According to Vitruvius (2.3.3), the Romans preferred the Lydion. In our database, almost all contexts with bricks having the attribute value Lydion are found in present-day Italy, with many located in the north. The other size categories are found all around Hellenistic Europe, even if their centres of gravity are located more to the south east than that of Lydion.

If these regional traditions alone would account for the significant deviations from randomness in Fig. 11, there would be no such deviations if we, in the randomisation, let contexts with a specific size category move to another site where there is, originally, a context with the same size category. Along this line, we considered the family of networks where all connected contexts must have the value Lydion of the attribute Size category, apart from the conditions used to define the default family of networks (Fig. 7). We chose Lydion since it is the most geographically clustered attribute value.

The resulting median edge lengths were shorter than those expected by chance (p = 0.14 for X = 9). The low significance can be explained by the fact that the ‘Lydion network’ only contains 33 sites, making the statistics poor. This weak indication that there are causal relations also within the small Lydion network was strengthened by the fact that the median edge length decreased steadily as X increased, in the same manner as in the network as a whole (Fig. 11).

Are All Fired Bricks Part of the Diffusion?

This visual test—to see whether more similar contexts tend to be closer than less similar contexts—was used to look for types of brick contexts that show no sign of being part of any diffusion. If we let the similarity level X increase in the subset of such contexts, the median edge length should not decrease. To be able to come to this conclusion, there has to be enough contexts of the given type so that the statistics are acceptable, as indicated by the clarity of the trend.

For almost all subgroups of attribute values that we used to define specific brick context types, there was a more or less clear trend that the median edge length indeed became shorter as X increased. The exceptions were brick contexts where the value of Structural use was Pavement and those where the value of the attribute Context was Sacred.

The indication that fired bricks used for pavement were not part of the same Hellenistic diffusion process as fired bricks used for masonry is not altogether surprising. Bricks used for pavement may be more linked to the use of roof tiles, which was widespread already in the classical period. It is seen in Fig. 5 that the first context in our catalogue with bricks used for pavement appeared already around 500 BCE.

Figure 12 shows characteristics of the family of networks defined by the requirement that all contexts have the value Pavement of the attribute Structural use, together with the default requirements of overlapping dating and at least X common attribute values. In all, there are 24 contexts at 17 sites in the database with the value Pavement. Figure 12a shows the network for X = 6 and illustrates the fact that these contexts are located at either side of the Ionian Sea, with an outlier context located in Karnak, Egypt. Figure 12b shows that the median edge lengths in these networks do not deviate significantly from those expected by chance, and that they do not decrease as X increases, as expected if the edges are the result of a diffusion process. Basically, contexts at the same side of the Ionian Sea are no more similar than those at opposite sides of the sea. Of course, these tentative results do not prove that fired bricks used for pavement were not part of the overall diffusion of fired bricks. All we can say is that there is no statistical sign of diffusion of such bricks in the material at hand.
Fig. 12

Characteristics of networks where two contexts are connected if and only if both contain bricks used for pavement, apart from the default conditions for connection used in Fig. 7. a The resulting network at similarity level X = 6, illustrating the location of the 17 sites where pavement bricks are found. b Median edge lengths as functions of X (Compare Fig. 11). Solid curve with circles: medians in the true networks. Solid curve: mean median edge lengths in randomised networks where the contexts are relocated randomly among the 17 sites. Dashed curves: median edge lengths that correspond to deviation from randomness at significance level p = 0.05. The irregular shape of the curves is the result of a small sample of sites with peculiar spatial distribution. There are no edges from one site to another for X > 8

While historical knowledge gives some credibility to the statistical indication that bricks used for pavement were not part of the diffusion process, this is not the case for the same indication targeting bricks used in sacred contexts. There is no obvious reason why sacred brick constructions should be causally separated from other brick constructions. Most probably, the indication of no diffusion is just a statistical fluke in a small data sample. There are only 21 sacred contexts at 16 sites in the catalogue. However, it must also be considered that the categorisation of these contexts as ‘sacred’ may not be causally relevant from a diffusion point of view. If the contexts in question would correspond to a randomly defined sub-group of public contexts, for example, the statistical evidence we have for diffusion among the public contexts is expected to be less clearly visible in the sacred subgroup.

The Distribution of Edge Lengths

It may be interesting to study the entire distribution of edge lengths, not just the median. As far as the similarity networks shadow the diffusion network (Fig. 2), this may tell us something about the structure of the social or societal network in Hellenistic Europe. The structure is lattice-like if almost all edge lengths are short, whereas a substantial amount of long edges suggest a small-world or a scale-free network (Fig. 3). A truly random network where all edge lengths are equally probable is strongly disfavoured by the above statistical analysis (Fig. 3b).

Figure 13a shows histograms of edge lengths in the default family of networks (Fig. 7). What is shown is the number of realised edges in the network at hand with lengths L within the bin [L,L+dL), divided by the number of possible edges with lengths within the same bin, given the geographical distribution of sites. This realisation ratio is smaller for long-edge lengths than for short ones, generally speaking, showing again that short-edge lengths are preferred in the similarity networks. In contrast, in the randomised similarity networks, where the locations of the sites are randomly permuted (Fig. 11), all possible edge lengths are equally probable, which would produce an approximately flat, horizontal histogram (not shown).
Fig. 13

The distribution of edge lengths L in the default family of networks (Fig. 7). a Histograms of the number of edges of a given length divided by the number of possible edges with this length. Bin width 30 km. b Cumulative distributions based on histograms like those in a. Bin width 10 km. The dashed line gives the expected distribution when all possible edge lengths are equally probable. c Log–log plot of the distributions in panel b. The scale in both the x- and y-axis is logarithmic. Compare Fig. 14

As the similarity level X increases, the realisation ratio in each edge length bin decreases. More interestingly, the realisation ratio of long edge lengths decreases more than that of short-edge lengths. This behaviour accounts for the fact that the median edge length decreases as X increases (Fig. 11).

It is hard to extract more information directly from these histograms. They are so non-smooth that no simple curve can be fitted to them. The standard trick in this situation is to study cumulative distributions. Figure 13b shows the sum of the realisation ratios in all bins with edge lengths equal to or larger than L. The distributions are normalised, that is, they are divided by the sum of all realisation ratios in all edge length bins. Therefore, all distributions start at unity on the y-axis for the smallest edge lengths.

Even though these curves are smoother than the histograms, it is still hard to say anything specific about their functional form. The exception is the dashed, bold curve that represents the expected distribution for a randomised network. This distribution is the same as that for a fully connected network, in which all possible edges are realised. For edge lengths up to 2,000 km, this distribution is very close to a straight line. This means that all these edge lengths are equally probable. The deviation from the straight line for very large edge lengths arises because there are few possible edges like that, and their lengths depend on the details of the geographical distribution of sites with brick contexts. The length of the longest possible edge is about 4,000 km, stretching all the way from Forum Iulii in France to Aï Khanoum in Afghanistan.

Figure 13c shows a log–log plot of the cumulative distributions in panel b. Now some interesting observations can be made. The distribution for the critical network for X = 9 seems to consist of two parts. It falls off slowly for edge lengths up to 200–300 km. Then it falls off more rapidly along an approximately straight line with slope –1. The change of behaviour around 200–300 km is seen also in the distributions for X > 9. Since large similarity levels X ≥ 9 in most cases are expected to reflect true causal links between contexts, this suggests that the spread of bricks between sites closer than 200–300 km belonged to one, common type of event, and that the spread between sites farther apart occurred, but belonged to another, rare type of event.

One may speculate that the two event types correspond to different kinds of social contacts or different diffusion processes, or both. One way of categorising social contacts is to distinguish between those made among ‘elite’ and ‘non-elite’ individuals, respectively, whereas two general types of diffusion processes can be described as ‘exchange of ideas and information’ and ‘transferral of skills and practices’ (e.g. professional training). In many cases, these alternatives overlap as they correspond to two important actor groups, commissioners (usually lay people) and builders. In the first case, the basic scenario would be that an elite individual commissioning a building is influenced by his or her peers when making decisions concerning the project. In the second case, architects or craftsmen transmit knowledge and skills amongst each other. Of course, there may be other scenarios as well (commissioners who are neither lay people nor elite; professionals who influence each other without transferring skills; contacts between builders and commissioners). It may be assumed that elite individuals managed to maintain social contacts over longer distances than non-elite individuals did, although most contacts probably were short (Östborn and Gerding forthcoming). In any case, the particular length 200–300 km must arise from the diffusion process in one way or another, since it is not ‘hardwired’ in the geographical distribution of sites. The distribution for randomised networks falls off smoothly.

The same kind of bipartite distributions arises in a wide range of networks, where additional conditions for connection of two contexts are added, apart from the basic requirement of a smallest general similarity level X. It seems robust. Regardless of the additional network conditions, the straight line with slope −1 always arises for X = 9. Figure 14 shows the almost identical distributions of two such networks. An attempt is made to identify the dividing line between short- and long-range behaviour more exactly. It seems to occur around 250 km.
Fig. 14

Log–log plot of cumulative edge length distributions like those in Fig. 13. Networks with the required similarity level X = 9 seem to have very similar distributions regardless the choice of additional conditions for connection. One curve corresponds to the additional condition of overlapping dating. The other curve corresponds to the same condition together with the condition that the phase should be defined for any pair of connected contexts and that their phase difference should not be greater than 50 years. The behaviour is different for short- and long-edge lengths. The distributions fall off rapidly for edge lengths above L ≈ 250 km. Above this length the functional form of the distribution is F(L)∼L−1

The straight line with slope −1 for long edges in the log–log plot means that the cumulative edge length distribution F(L) for L > 300 km follows a power law F(L)∼L−1 and that the density function F(L), approximated by histograms such as those in Fig. 13, follows a power law F(L) ∝ L−2.

These are nice properties, but we should be careful not to conclude that they imply that the probability that fired bricks spread across distance L was proportional to L−2, that the influence of a brick site weakened with distance just like the gravitational force of a massive body. Even though we have highlighted the critical similarity level X = 9, we cannot be sure that it gives rise to optimal similarity networks that mimic the diffusion network as well as possible (Fig. 2). For X > 9 the distributions fall off faster, and we cannot claim with confidence that they follow a power law (Fig. 13c).

The Degree Distribution

The degree distribution of the similarity networks may tell us something about the hierarchical structure of the societal network in which the diffusion took place. If similarity networks that mimic the diffusion network reasonably well are scale-free, there must have been some societal mechanism that favoured large degrees in the diffusion network. A few dominant hubs in the societal network must have existed, from which the spread of influences such as innovations were governed.

Figure 15 shows degree distributions from the default family of networks (Fig. 7). Naturally, as the required similarity level X increases, the number of nodes with large degree decreases. The histograms in Fig. 15a are irregular and hard to analyse. To be able to extract more information, cumulative distributions were plotted, just as for the edge length distributions in Fig. 13. In Fig. 15b, the height of the curve at degree D gives the proportion of all nodes that have degree equal to or larger than D. (The isolated nodes with degree zero are excluded in the counting.)
Fig. 15

Degree distributions in the default family of networks (Fig. 7). a Histograms of the number of nodes with a given degree D. b Cumulative distributions based on the histograms, giving the number of nodes with degree equal to or greater than D. c Semi-log plot of the distributions in panel b. The y-axis is logarithmic but not the x-axis

Figure 15c shows a semi-log plot of the cumulative distributions in Fig. 15b. In such plots, a straight line with a negative slope −c corresponds to an exponential cumulative distribution F(D)∼ecD, and a density function of the same form F(D)∼ecD. We see that all degree distributions fall off faster than such an exponential distribution. Other network families displayed the same behaviour.

Therefore, we are confident that no similarity network is scale-free. This would have meant that the cumulative distribution (and the density function) was given by a power-law F(D)∼D-a for fairly large degrees D. Such distributions fall off slower than the exponential. We would have seen a convex curve in the semi-log plot among all the concave ones. In other words, power law distributions have ‘fatter tails’ than exponential (or normal) distributions. Compare, for instance, the power law edge length distribution for X = 9 in Fig. 13b with the corresponding degree distribution in Fig. 15b.

The fact that the distributions fall off faster than the exponential means that they may be approximately normal, at least for large degrees. However, the statistics are too poor to make any definitive conclusions. Close to normal distributions suggest a more ‘democratic’ diffusion process.

Consider, for instance, the following model. The innovation first appears at a random site. It spreads to another site. After that, all sites where the innovation is already adopted have equal probability of transferring the innovation to a new site. In this process, all sites are equivalent, and the degree distribution is approximately normal for large degrees.

One should be careful, however, not to conclude that a close to normal degree distribution in a diffusion network means that the overall society is decentralised. Such a conclusion is only valid under the assumption that the edges in the diffusion network are a representative sample of the edges in the societal network (Fig. 2). This is not the case if the probability of adoption is smaller in societal hubs than at more peripheral sites. In our case, this is a real possibility, suggested by the fact that fired bricks did not reach Athens or Rome until the last decades of the first century BCE. This behaviour is possible to simulate in simple diffusion models (Watts 2002) and is explored in a study where the diffusion of fired bricks is modelled (Östborn and Gerding forthcoming).

Network Structure

The above statistical analyses have made us confident that one or more causal processes were responsible for the spread of fired bricks across Hellenistic Europe. It is therefore of interest to try to infer the structure of the diffusion network and, indirectly, the structure of the societal network in which the diffusion took place.

Since we were confident that the brick contexts did not appear randomly and independently, we expected that the structure of the similarity networks we used as proxies for the diffusion network deviated significantly from that of random networks. The interesting question was in what way it deviated from randomness. To this end, structural quantities of the default family of similarity networks (Fig. 7) were compared with those of networks belonging to the same family, but for which the attributes were randomly permuted (Östborn and Gerding 2014; Fig. 8). Since our database has hierarchical elements, we had to make sure that the attribute values of each context were mutually compatible in the randomised networks (Östborn and Gerding 2014).

Figure 16 shows eight structural quantities in the true and in the randomised networks. The default similarity conditions were used (Fig. 7), but the characteristics that we highlight were robust to changes in these. Two thousand random permutations were performed before the averages of the relevant quantities were computed, and the deviations that correspond to significance levels p = 0.05 were calculated. It was checked that the number of permutations was large enough to permit stable results.
Fig. 16

Structural quantities as functions of the similarity level X in the default family of network conditions (Fig. 7). These are applied to the true database and to randomised databases. The solid curves with circles correspond to the true database, and the solid curves correspond to the averages of the randomised databases. The dashed curves are the deviations from the expected value in a random database at significance level p = 0.05. The network size is defined as the number of nodes with degree larger than zero

Figure 16a shows the number of nodes with degree larger than zero as a function of the similarity level X. The number of these non-isolated nodes can be used as a measure of the network size. The size of the true network is larger than that of random networks. For X > 6, the deviation from randomness is highly significant.

Figure 16b shows the clustering coefficient. This seems to be the crucial quantity to pinpoint the structural difference between the true network and the random networks. The clustering coefficient is much higher in the true network. This explains the difference in network size. It is harder to isolate nodes in a highly clustered network since most of them belong to tightly interconnected groups.

The deviation from randomness is extremely significant, except for X = 10. At this significance level, the random networks are very small, and the high variance of the clustering coefficient reflects the fact that the absence or presence of a single edge may change the clustering coefficient substantially. In a network of just three connected nodes, an edge that closes the triangle makes the clustering coefficient jump from zero to one.

The crucial difference between the true and randomised clustering is that the expected value of the latter decreases steadily towards zero as X increases, whereas the former stays high. It even starts to increase around X = 6. How does this come about? It happens because in the true database there are clusters of very similar contexts that are clearly separated from each other in terms of similarity.

Consider Fig. 17. The three contexts in cluster 1 have almost the same attribute values and so have those in cluster 2. But contexts in cluster 1 have few attribute values in common with contexts in cluster 2. Let X common attribute values be the only requirement for connection. At X = 0, all contexts are connected to all the others. All triplets are closed into triangles; there are no edges missing. The clustering coefficient is one. As X increases, more and more of the dashed edges that connect contexts in different clusters disappear. The clustering coefficient decreases since some triangles lose one of their legs. But at some similarity level X2, there are no longer any dashed edges. But the contexts within each cluster are so similar that they are still fully connected internally. The clustering coefficient again rises to one.
Fig. 17

The clustering coefficient in a network that consists of two clusters that are clearly separated in terms of similarity. The dashed edges are weak in the sense that they disappear before the solid edges when the required similarity level X increases. The clustering coefficient first decreases, but recovers when all the dashed edges are gone

This is, of course, an idealised situation. But the moral remains: As the similarity level increases, islands of similar clusters are panned out and become visible. In a random database, the attribute values vary more continuously in the population, less stepwise, without distinct clusters of similar contexts. There can be no effect such as that in Fig. 17. Therefore, the clustering coefficient decreases steadily as X increases.

The deviation from randomness is less dramatic when it comes to the average shortest path length and the diameter (Fig. 16c and d). Just as these quantities peak at the critical similarity level X = 9 in the true network, they peak around X = 8 and X = 9 in the random networks. The random networks also possess a critical similarity level above which a single dominant component fragments. However, above criticality the behaviour differs significantly from that of the true networks. For X > 9, the latter have much longer average shortest path length and diameter. This reflects the fact that in the random networks, the dominant component immediately fragments into small pieces. In the true networks, it splits into two main components, as we have seen above. This in turn reflects the fact that there are significantly more distinct clusters of similar contexts in the true networks, as concluded from the behaviour of the clustering coefficient.

Figures 16e and f confirm these findings. It is seen that the size of the largest and second largest components are significantly larger in the true networks close to and above criticality. The high peak in the size of the second largest cluster (Fig. 16f) is the result of the split into two main components. The peak exists in the random networks also but is much lower. When the network conditions are varied, the difference often becomes even more marked, as the size of the second largest component in the true networks often reach 40–50 nodes. (We define component size as the number of member nodes.) Apparently, the true networks not only contain more clearly defined clusters than expected by chance, but also contain clusters of more varying sizes, both small and large ones.

The average degree in the networks decreases steadily as X increases, as well as the variance among the nodes of the degree. This holds true both for the true and for the random networks. For a given X, both quantities are, however, significantly larger in the true networks, at least for X > 6. Again, it is possible to interpret this difference in terms of the qualitatively different behaviour of the clustering coefficient. The existence of clusters in the true networks means that the average degree rises since they become very well connected internally; edges tend to bridge any given context to most others within the cluster. On the other hand, there will also be loners that fall between the clusters, and sometimes act as bridges between different clusters, in terms of similarity. These tend to have few connections. The contrast between the two types of nodes makes the variance rise.

One may think of another, more trivial explanation. We observed in Fig. 16a that for a given X, the size of a true network is larger than that of a random network. The more nodes in a network, the more edges are possible. This may increase the average degree. If the average of a quantity increases, the variance tends to follow. To check whether this may explain the differences between the true and random networks, the average degree and its standard deviation was divided by the network size (Fig. 16g and h). The differences did not go away and were particularly significant in the most interesting critical region around X = 9.

Similar checks were made for the other structural quantities. They were normalised, not just with respect to network size in terms of the number of nodes, but also with respect to the number of edges. In neither case was it possible to make the curves that describe the true networks fall within the region allowed by the assumption of randomness (dashed curves). Furthermore, it was impossible to do the trick by re-scaling X, to say that the behaviour of the true network at similarity level X corresponds reasonably well to the behaviour of the random networks at level f(X) for some function f. This shows that the structure of the true similarity networks not only differs quantitatively from that suggested by the assumption of randomness, but also differs qualitatively. As usual, varying the network conditions away from the default network family did not change the main findings.

In conclusion, the crucial insight provided by the structural network analysis is that the way fired bricks were used was much more clustered than suggested by chance. In other words, there were ‘fired brick cultures’ in Hellenistic Europe. More than that, there were such cultural clusters of many sizes, just like there are countries of many sizes. The analogy is not perfect, since brick clusters are defined in terms of attribute similarity and are not always geographically focused. The two largest clusters have been discussed at length above, those that correspond to the early network component in the south east, and the late component in the northwest (Fig. 9). The new finding here is that it is extremely unlikely that such large clusters would appear by chance.

Within each large cluster one may identify subclusters with even higher degree of similarity. Their existence is demonstrated by the fact that the second largest component in the true network family remains very much larger than expected by chance for all X > 9 (Fig. 16f). This network structure of small and large clusters, and clusters within clusters, is reminiscent of fractals, where new patterns appear at all scales. Apparently, the diffusion of fired bricks across Hellenistic Europe did not occur as a simple wave rolling forward, neither geographically nor in terms of brick attributes. This complex evolution may be connected to the long time scale of the process, which allowed the use of fired bricks to ‘settle down’ and form cultural clusters along its path from the north Aegean to Rome.

Relative Importance of Attributes in the Diffusion Process

We may ask what attributes were important in the diffusion in the sense that they inspired adopters and therefore were mimicked by them when the use of fired bricks was transferred from one site to the next. To answer this question, we may look at attributes that often have the same value in a pair of similar contexts. Such an investigation is relative: These attribute values will be left unchanged in a transfer more often than others. In terms of clusters of similar, related contexts, such attributes can be seen as the cement that keeps the clusters together.

However, some attributes have the same value in almost all contexts and will therefore trivially be judged important in the above sense. For instance, this goes for Poorly fired, which almost always has the value No, and Structural use, which most often has the value Masonry. Such background attribute values are not very interesting: What we look for is crucial attributes that tend to have the same value in similar contexts, but only in these.

We addressed the problem in the following way. For each attribute Y, we defined the network family in which a pair of contexts is connected if and only if their dating overlaps, they have at least X attributes in common, and attribute Y is the same. We counted the number of edges N(X, Y) that were present in these networks. Then we computed P(X, Y) = N(X, Y)/N(1, Y). The quantity P(X, Y) is the probability that a randomly chosen pair of contexts that have attribute Y in common fulfil similarity level X, given that they have the potential to be causally linked (that their dating overlaps).

As usual, we focussed on the critical similarity level X = 9, judging that this similarity network resembles the diffusion network the most (Fig. 2). We interpreted a high value of P (9, Y) to be a sign that attribute Y was important in the diffusion. According to this criterion, the 13 attributes that describe the context per se were ordered as follows, with the most important attribute first: (1) Structure, (2) Function, (3) Binding, (4) Size category, (5) Context, (6) Thickness, (7) Plaster, (8) Shape, (9) Combined with mud brick, (10) Stamp, (11) Structural use, (12) Poorly fired, and (13) Tile bricks. To be able to visualise the results clearly (Fig. 18), the relative importance of each attribute Y compared with the median attribute Plaster was calculated as R(X, Y) = P(X, Y)/P(X, Plaster).
Fig. 18

Relative importance of attributes in the diffusion. Attributes are judged to be important if their values are often shared by two similar contexts, but less often so by two dissimilar contexts. See main text for further explanation

The three attributes Structure, Function and Binding stand out at similarity level X = 9. Structure and Binding are technical attributes (Table 1), and this finding may be seen as a sign that craftsmen were often involved in the transfer of fried bricks from site A to site B. Building methods may either have been learned by the local craftsmen at site B from those at site A, or skilled craftsmen may have been recruited from site A to construct a brick building at site B. The alternative, judged to be less likely in our analysis, is that decision-makers at site B were inspired by some building at A (say a city wall) and ordered the local craftsmen to erect a similar building, solving the technical problems on their own.

All of the top three attributes concern building elements, as contrasted with Context and Structural use, which concern overall features. The latter two attributes are judged to be less important (Fig. 18). Assuming that elite commissioners are more interested in the general picture, this fact again suggests that non-elite craftsmen and builders were most often crucial in the transfer of fired bricks.

At even higher similarity levels than X = 9, Size category races to the top of the list (Fig. 18). This reflects the fact that at these similarity levels, the only remaining interconnected contexts are located in Italy, where the size category Lydion dominated. This may or may not be significant for the diffusion process, since Lydion and the other size categories not only reflect regional traditions in the use of fired bricks, but also in other building materials such as mud bricks. It may thus be a confounding covariate that does not indicate a causal relationship, as discussed above.

Diffusion Directions

Just as we tested whether the lengths of the edges in the similarity networks differ significantly from those expected by chance, it may be interesting to see whether the edges have a preferred direction. This would indicate a diffusion direction, not for individual transfers of fired bricks from site to site, but for the process as a whole. To perform such a test, we again used one of the random permutation tests outlined in Östborn and Gerding (2014), Fig. 9.

Two sets of temporal phases were defined, one early and one late set; let us call them S1 and S2, respectively. For instance, we might be interested in the direction in which the centre of gravity of brick usage moved from phase 175 BCE to phase 150 BCE. Then S1 = {175} and S2 = {150}. We may also be interested in the diffusion direction at a larger temporal scale, comparing the third century BCE with the second century. Then, we choose S1 = {300,275,250,225,200} and S2 = {200,175,150,125,100}.

The next step was to draw a directed edge from each of the contexts with a phase that belongs to S1 to all the contexts with a phase that belongs to S2. After normalisation to unit length, all these vectors were summed vectorially. The resultant provided a measure of the collective diffusion direction.

To test whether the existence of a diffusion direction was statistically significant, the locations of all contexts in the database were randomly permuted in the same manner as in the test for edge lengths. Five thousand random permutations were performed.

Figure 19 shows those time intervals during which there was a significant diffusion direction (p < 0.05). Each interval was divided in the middle in one early set S1 and one late set S2 of phases, as described above. All intervals that belonged to the period 400–1 BCE and contained two, five, nine or 17 consecutive phases were tested.
Fig. 19

Significant diffusion directions (p = 0.005) during the marked time intervals. The significance level is noted to the right in each interval. The length of the arrow is proportional to the length of resultant obtained when all individual direction indicator vectors are added. It is thus a measure of the spatial clarity of the trend

During the fourth and third centuries BCE, there was a weak but significant trend that the use of fired bricks diffused to the south west. During the phases 175 and 150 BCE, there was a sudden, clear jump to the northwest. This is the only short time interval, containing just two phases, in which the diffusion direction was significant. The diffusion towards the northwest continued successively until the last decades BCE. The motion was less clear than during the initial jump, but more clear than the diffusion to the south west during the first half of the period of investigation. Looking at the entire period 400–1 BCE, there was an overall trend that fired bricks diffused towards the northwest.

Regarding the fact that only one of 16 diffusion directions during the shortest time intervals (two consecutive phases) was significant, it should be noted that this is close to the expected outcome from 16 random samples with no directionality whatsoever. The distinct jump to the northwest from the phase 175 BCE to the phase 150 BCE can nevertheless be worth highlighting if there are few unknown brick contexts from this time period. Then the result should be seen not as a statistical outcome, but as a characterisation of the actual use of fired bricks during this period. Unfortunately, we cannot say with any certainty that this is the case. However, we can justify the jump as the beginning of a longer trend of diffusion towards northwest, a trend that is very significant viewed at larger temporal scales. Intuitively speaking, the beginning of this trend can borrow statistical significance from the events that followed after it.

The findings visualised in Fig. 19 go together well with the study of shortest paths between the earliest and the latest brick contexts (Fig. 8). These paths first moves from the North Aegean south west to southernmost Italy, then turn northwards, more or less along the northwestern axis defined by the Italian peninsula, sometimes jumping back and forth along this axis. The clear jump towards the northwest during the phases 175 and 150 BCE may be seen as the ignition of the second stage of the brick diffusion that eventually created the late northwest network component discussed in connection to Fig. 9.

Conclusions

In this paper, we have tried to extract as much historical insight as possible from similarity network analysis of a database of Hellenistic fired brick contexts. Here we summarise our findings so that the emerging historical picture stands out clearly. We also judge which of the hypotheses could have been put forward without network analysis, to give a feeling for the ‘net benefit’ of the approach.

There was one or several causally linked diffusion processes in which the use of fired bricks spread across Hellenistic Europe. This was not clear to us to begin with, given the apparent randomness of the material. However, signs of ‘micro diffusion’ within small parts of the data could be seen by visual inspection. At the same time, we were aware of the danger of fooling oneself and finding meaning in random patterns.

Most transfers of fired bricks took place between nearby sites, but there were also long-range social contacts that sometimes caused a transfer from one site to another faraway site. The existence of very similar far apart contexts made us think along these lines before we performed any network analysis. But the statistical network analysis placed the hypothesis on more solid ground.

The short- and long-hand brick transfers had different statistical characteristics, suggesting that they were the result of different social processes. Presumably, short-hand transfers took place among craftsmen, whereas long-hand transfers took place among decision-makers higher up in the social hierarchy. The watershed between the short- and long-range transfer regimes took place around 250 km. We could never have come to this conclusion without careful analysis of the edge length distribution of a variety of similarity networks.

The technical brick attributes, such as structure and binding, were the most crucial ones in the diffusion, strengthening the idea that skilled craftsmen played an important role in the social network through which fired bricks spread. This is very hard to see in the material without the use of network analysis.

The diffusion of fired bricks was not governed by a few dominant sites, but occurred between sites of more or less equal importance. The social network in which the diffusion took place was probably a small world, but not scale-free. We noted early on that brick contexts in Rome and Athens were remarkably absent in the database before fired bricks reached Rome at the end of the first century BCE. Since these cities are the largest potential hubs of influence, the idea emerged that the diffusion of Hellenistic fired bricks took place in a more democratic fashion between smaller sites. However, to describe the situation in terms of small-world and scale-free networks, it was necessary to perform network analysis to match the outcome to the formal definitions of these concepts.

The diffusion of fired bricks can be divided in one early and one late phase. The early phase covered the fourth and third century BCE. During this period, the use of bricks diffused to the south-west from the North Aegean, quickly reaching Calabria and Sicily. Throughout the early phase, fired bricks were mainly concentrated in the southern and eastern parts of Hellenistic Europe. Even if fired bricks were used in a variety of ways, they were most established and homogeneous in sepulchral contexts. The geographical distribution of fired bricks during the early phase can, of course, be seen without network analysis. The direction of diffusion to the south west is discernible in the database, but network analysis was necessary to establish that the movement is statistically significant. It was very helpful to use cluster analysis based on similarity networks in order to identify the central role of the sepulchral contexts.

There was a critical period for the diffusion during the first half of the second century BCE during which fired bricks were rare. At the same time, their use quickly jumped northwards along the Italian peninsula. A group of sites along the coast of Albania may have played a crucial role as a pillar supporting this bridge towards the late phase of brick usage. The critical period can be seen in the temporal brick distribution (Fig. 5). The jump to the north is clearly visible in the database. It stands out in the statistical network analysis as the only significant short-term geographical movement. The possibly crucial role of present-day Albania could be guessed by comparing the spatial distribution of bricks and their dating, but the idea was strengthened by similarity network analysis.

During the late phase, fired bricks were most common in the northern and western parts of Hellenistic Europe. This phase lasted from the latter half of the second century BCE to the explosion of fired brick usage at the beginning of the Common Era. It was characterised by a higher degree of homogenisation in the way bricks were used. This is most clearly visible in large-scale public or military buildings, suggesting that the use of fired bricks became established at a higher societal level during the last century BCE. Just as for the early phase, the geographical brick distribution during the late phase can be seen without the use of network analysis. Such analysis was useful, however, to establish statistically that a successive diffusion of fired bricks to the northwest along the Italian peninsula was ongoing during the late phase. Network analysis was crucial in order to see that the overall degree of homogenisation was larger in the late phase. It was also very helpful to target public and military contexts, even if we had previously noted a group of very similar brick fortifications in northern Italy.

The early and late phases can be seen as two ‘cultural clusters’, meaning that within each phase the way fired bricks were used was quite homogeneous. There were several such cultural clusters of many different sizes, including these two large ones, just as there are countries of many different sizes, including a few superpowers. Network analysis was crucial for seeing that the early and late phases were not just temporally and geographically separate, but that they were also two cultural clusters. Statistical network analysis was necessary for seeing that there was much more clustering in the material as a whole than expected by chance.

To build walls with rectangular bricks remained the basic way to use fired bricks throughout the Hellenistic period, even though the characteristics of such brick walls changed with time. For instance, the bricks became successively thinner. The more unusual and specialised ways for using bricks probably emerged as smaller branches from this main stem of evolution more or less independently at different times and at different locations. That rectangular bricks in walls are basic is quite natural, and it is easy to see in the database that they tend to become thinner in the late Hellenistic period. It required network analysis, however, to interpret the diffusion process as a main stem of evolving wall bricks with smaller, independently emerging, branches of specialised bricks.

The network analysis gave more insight about the use and diffusion of fired Hellenistic bricks than we had expected from the beginning. It must be remembered, though, that hypotheses based on network analyses should be seen as preliminary. Naturally, to gain more understanding, the material should be put into an historical context. A database analysis, like the one provided here, is never enough. A broader analysis of Hellenistic fired bricks will be given elsewhere (Gerding and Östborn forthcoming).

Footnotes

  1. 1.

    More information about the program and the numerical analysis can be found at http://projekt.ht.lu.se/lateres

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Classical Archaeology and Ancient History, Department of Archaeology and Ancient HistoryLund UniversityLundSweden

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